mellowtigger

Entries tagged with math/physics

Would terraforming Mars be self-defeating? Many articles discuss how to terraform Mars, but they seem to assume that a human-friendly atmosphere would just "stay there". Mars is smaller than Earth, therefore its gravity is weaker, so would a new atmosphere just evaporate into space because the planet could not retain it gravitationally?

The short answer is: Mars can retain a human-friendly atmosphere.

The long answer is detailed below. I originally wrote this article in January 2006 while examining Earth. Since that website will disappear in coming months, I wanted to republish it here in my blog so it stays online somewhere. This version, obviously, will focus on Mars.

Retention of Atmosphere by a Planetary Body

Whether we consider small atoms or huge rocket ships, there is just one property that determines if something can escape the pull of the planet's gravity: velocity. If something travels with enough speed directly away from the planet, then it will be able to overcome the persistent force that would otherwise send it back to the surface. It doesn't matter if we measure baseballs or rockets or molecules of air. They all obey this same basic principle. If it moves fast enough, then it will escape.

Reaching escape velocity is helpful to us when we launch probes to other planets. We might worry, though, about unintended loss of material from a planet. After all, air itself can escape out into space! If a molecule of air is travelling fast enough, it will leave the atmosphere forever. Two questions become very important in understanding this effect:

1) how fast does the molecule have to move in order to reach escape velocity, and2) what can cause it to travel that fast?

The effect of gravity is well studied, and determining escape velocity is an easy calculation. The only information that we need about a planet is its mass and its radius. We insert those two values into a formula and then we find the speed needed to escape the surface of the planet while traveling straight upward.

name

value

units

escape velocity formula

velocity = √2 * G * mass / radius

m / s

gravitational constant

G = 6.67 x 10-14

Nm2/g2 = m3/(g * sec2)

mars mass

mass = 6.42 x 1026

g

mars radius

radius = 3.40 x 106

m

now just plug into the formula...

mars escape velocity =

√2 * G * mass / radius

√2 * (6.67 x 10-14) * (6.42 x 1026>) / (3.40 x 106)

√(m3/(g * sec2)) * (g) / (m)

√2 * (1.26 x 107)

√(m2/sec2)

5.02 x 103

m / s

So anything on Mars that moves straight upward at 5.02 kilometers per second will manage to escape the planet and travel on to other destinations. That speed is enormous! That's why space launch vehicles are so huge; they require large amounts of fuel to accelerate their payload to escape velocity. Could anything around us reach such speeds unassisted by human intervention? Could air itself travel that fast?

It is easy to measure the speed of a large object. For example, every car comes equipped with a speedometer that informs the driver of the car's rate of movement. It seems much more difficult to measure the speed of a very small object, though. How do you watch the movement of an atom, an object so small that you can't even see it? It turns out that we do it all the time. Rather than measure the speed of a single atom, though, we measure the average speed of lots of atoms together. It's simple, really. We just take its temperature.

A speedometer measures the movement rate of a large vehicle, and a thermometer measures the relative movement rate of lots of atoms together. The higher the temperature, the faster those atoms are moving. Likewise, the colder that something gets, the slower its atoms are moving around. The relationship between speed and temperature is more obvious in a gas, but even in solids it is still true that the individual atoms are jittering quickly in their place. The Kelvin temperature scale is anchored at zero degrees at the low end of the scale because that is the point at which the atomic movement has slowed as much as it can. It's not possible for the atoms to move any slower, so they are as cold as anything can get. Zero degrees Kelvin is called "absolute zero". The relationship between speed and temperature is easy to demonstrate.

Fill a balloon with air and tie off the end of the balloon so that no air escapes. Notice the size of the baloon. It keeps that size because of the air pressure inside it. That pressure is generated by all of the small molecules of air zooming off in random directions until they hit the wall of the balloon surface. They push outward and then bounce back to travel in another direction. They ricochet inside the balloon, exerting a constant pressure trying to expand the balloon outward. Similarly the air outside is bouncing against the balloon trying to crush it inward. They eventually reach equilibrium, but you can change that balance by changing the temperature in the balloon. Hold the balloon under hot tap water (or hold over a pot of boiling water). The faster molecules hit the wall of the balloon with greater force, expanding the balloon. Hold the balloon under cold tap water (or place in the refrigerator). The slower molecules hit the wall with less force, allowing the balloon to shrink. Temperature affects air pressure (but that's just a bonus lesson to be learned) because it affects the speed of molecules.

The law of temperature and pressure is also well studied, so to calculate the speed of an air molecule we can once again use a simple formula. All we need to know is the air molecule's mass and temperature.

The concept that we're examining is the "kinetic energy" (energy of motion) of molecules. Physics already has two simple formulas for measuring kinetic energy. If we compare them to each other, we can solve to find the velocity that we're needing.

kinetic energy(motion)

=

kinetic energy(average for a gas)

1/2 * (m * v2)

=

3/2 * (k * T)

m * v2

=

3 * k * T

v2

=

(3 * k * T) / m

v

=

√(3 * k * T) / m

The value of "k" is the Boltzmann constant, 1.38 x 10-23 Joule / Kelvin. The average temperature "T" on Mars is 220 Kelvin. (For the rest of the world, that's -53 Celsius. And for backwater Americans, that's -63 Farenheit.) And those are the last two values that we need. Let's fill in the formula and see what we find out.

name

mass(in kilograms)

formula

√(3 * k * T) / m =

average speed(in meters/second)

H, hydrogen atom

1.67 x 10-27

√(3 * (1.38 x 10-23) * (220)) /(1.67 x 10-27) =

2.34 x 103

H2, hydrogen gas

3.35 x 10-27

√(3 * (1.38 x 10-23) * (220)) /(3.35 x 10-27) =

1.65 x 103

He, helium atom

6.65 x 10-27

√(3 * (1.38 x 10-23) * (220)) /(6.65 x 10-27) =

1.17 x 103

C, carbon atom

1.99 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(1.99 x 10-26) =

6.77 x 102

N, nitrogen atom

2.33 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(2.33 x 10-26) =

6.25 x 102

O, oxygen atom

2.66 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(2.66 x 10-26) =

5.85 x 102

H2O, water

2.99 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(2.99 x 10-26) =

5.52 x 102

CO, carbon monoxide

4.65 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(4.65 x 10-26) =

4.43 x 102

N2, nitrogen gas

4.65 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(4.65 x 10-26) =

4.43 x 102

NH3, ammonia

5.15 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(5.15 x 10-26) =

4.21 x 102

O2, oxygen gas

5.31 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(5.31 x 10-26) =

4.14 x 102

CO2, carbon dioxide

7.31 x 10-26

√(3 * (1.38 x 10-23) * (220)) /(7.31 x 10-26) =

3.53 x 102

According to these results, none of the atoms or molecules listed here could reach escape velocity on mars because they are all below the necessary 5.02 x 103 m/s. That's good news, right? Well, not really. The problem arises from the fact that we measure the temperature of a large sample of air all at once, not each individual molecule separately. The velocity that we compute is just the average velocity of all molecules in the group of air that we measured; it is not the velocity of any particular molecule. Some of them are moving faster than this calculated amount, and some are moving slower.

Most molecules will have a velocity near the average in their group, but a few will have twice that velocity. Fewer will have four times that velocity. Even fewer will have six times that velocity. The general rule (for which I have not yet found a mathematical explanation) is that a molecule will escape into space if 6X its average velocity is sufficient to reach the planet's escape velocity. Once that threshold is reached, the planet cannot retain that type of molecule because random exchanges of energy will give too many individual molecules the boost of kinetic energy that they need to speed away from the planet.

It's time for the big question... so how does Mars fare under this new scheme?

name

average speed(in meters/second)

6X average speed(in meters/second)

> escape speed?(5.02 x 103 m/s)

H, hydrogen atom

2.34 x 103

1.40 x 104

279%

H2, hydrogen gas

1.65 x 103

9.89 x 103

197%

He, helium atom

1.17 x 103

7.02 x 103

140%

C, carbon atom

6.77 x 102

4.06 x 103

81%

N, nitrogen atom

6.25 x 102

3.75 x 103

75%

O, oxygen atom

5.85 x 102

3.51 x 103

70%

H2O, water

5.52 x 102

3.31 x 103

66%

CO, carbon monoxide

4.43 x 102

2.66 x 103

53%

N2, nitrogen gas

4.43 x 102

2.66 x 103

53%

NH3, ammonia

4.21 x 102

2.52 x 103

50%

O2, oxygen gas

4.14 x 102

2.49 x 103

50%

CO2, carbon dioxide

3.53 x 102

2.12 x 103

42%

By these calculations, Mars retains all but hydrogen (whether in its atomic or molecular form) and helium. So if we found a way to mimic Earth air composition on Mars, the planet could gravitationally retain the atoms and molecules. But there's still a catch...

The mean temperature on Mars is 220 Kelvin. What if we wanted to improve the temperature as well as the chemical composition of the atmosphere? Would the air at 282 Kelvin (Earth's current average) still remain bound to the planet? Plugging the new temperature into the formulas, I find very similar results. The next heaviest common atom is Carbon, and it reaches only 92% of the necessary velocity to escape. Since it doesn't usually exist in a gaseous state, we have no worries about its approach to 100%. Nitrogen reaches 85% which might be troublesome, depending on just how long a time period we want to examine, or how accurate is the "6X" rule of thumb. If "6X" is just guesswork, then Nitrogen retention might be problematic on Mars.

Atomic and molecular hydrogen reach either double or triple the needed velocity to escape Mars. They will leave "quickly", so it's important that hydrogen is always bound together in heavier molecules or is replenished regularly. Of course, there is the ongoing problem of natural geologic processes that will affect the air composition, but we'll assume that the terraforming plan takes them into account.

Future exercise: What is the rate of hydrogen loss on Mars? How much hydrogen and helium does our sun deposit on Mars via the solar wind? Would it offset the natural gravitational losses?

Disclaimer: I am not responsible for errors caused by variation in temperature due to global warming, solar nova, or universal collapse. The rest of it is definitely my fault though.

Sources:

"Universe", 6th ed, by Roger A Freedman and William J Kaufmann III, 2002, ISBN 0-7167-4647-6, for formulas and concepts

A good song for ending this day. Obama has only 257 days remaining to announce the extraterrestrial inhabitants already on our planet. (Everything that I read on the internet is always true, of course.)

I watched the movie "Contact" for the umpteenth time tonight. I still like it a lot, as much as I liked Carl Sagan's book. Few movies accomplish that feat. I even liked it better than the new Doctor Who episode. I was flipping channels to watch both, and I found myself pausing Doctor Who so I could spend more time watching Contact. The television series is supposed to inspire hope and optimism, but I got more of that experience from the old movie tonight. Here, that optimism spills over into this song too. "We'd like to make a contact with you, baby."

On a related note, physicists have been able to produce hexagonal currents in a fluid. (Image on the left.) It has something to do with differential rotation, meaning that some parts spin faster than other parts. Instead of the typical whirlpools that I'd expect to see, they managed to get hexagons. The news article includes video footage of the effect which is also cool stuff.

This discovery is important, because it means that we don't need to invent hyperdimensional physics in order to explain the same effect on Saturn's north pole. (Image on the right.) There was already an alternative sonic explanation too. The important thing, though, is that Occam's razor allows us to leave the pan-dimensional mice out of the explanation now. We can recreate Saturn's hexagon in a lab, so it loses its mystique.

There was a flap in the news last year about the new super collider producing black holes. When they're small, though, black holes are supposed to evaporate faster than they expand. As they evaporate, they release energy. This is the radiation that made Hawking famous. So what if you were able to harness that radiation for useful purpose? Somebody did the calculations. :)

"Using the formulae from the section above, we find that a black hole with a radius of a few attometers at least roughly meets the list of criteria (see Appendix). Such BHs would have mass of the order of 1,000,000 tonnes, and lifetimes ranging from decades to centuries. A high-efficiency square solar panel a few hundred km on each side, in a circular orbit about the sun at a distance of 1,000,000 km, would absorb enough energy in a year to produce one such BH."- http://arxiv.org/pdf/0908.1803v1

In other words:

Build a massive solar cell array in space.

Collect the charge for a long time, a year or more.

Discharge the energy in one enormous flash through a sphere of gamma ray lasers.

Their beams converge at a single point to produce a small black hole with the mass of a million tons. (The reverse usage of the E=mc2 reaction that produces nuclear bomb explosions.)

Transport the black hole to your starship.

Siphon the evaporation energy until the black hole finally disappears.

Alternatively, feed it mass to restore its potential energy.

Repeat as needed.

Their paper goes on to discuss the feasibility of producing the black hole, of producing the starship drive, of harnessing the black hole in a power plant, etc. Besides finding it feasible, they think it could surely be improved.

"A BH with a life span on the order of a century would emit enough energy to accelerate itself to relativistic velocity in a period of decades. If we could let it get smaller and hotter before feeding matter into it, we could get a better performance."

Moreover, they say that it's perfectly reasonable to think that other star-faring civilizations are already using such technology. Because this technology "would emit gravitational radiation at nuclear frequencies", they say that SETI projects should consider building new detectors that work in this range. We might be able to detect galactic neighbors already using such energy devices. (Once somebody on Earth builds any kind of gravity detector that works.)

With the equivalent of a minimum wage job, it's going to take a while to get myself out of debt and save up money for classes. So what to do in the meantime? I've been pondering two things. A) Programming a game. (a decades-old interest of mine) B) Defining division by zero.

Division by zero is not "impossible", as I understand it, because it's merely "undefined" at the moment. That makes it an attractive topic of interest. A place to leave an intellectual mark, so to speak, by solving the problem. Here follows my attempts to find appropriate metaphors for what I imagine is division by zero.

I think the solution has something to do with rotation. When I imagine division, I end up "turning" my perspective. Division by zero is a really intensely fast rotation... so fast that my orientation is lost altogether. Is it possible to measure anything (scalar) without even a single dimension for orientation? Some information is lost but not all of it. Division by zero has a "place" (in my mind) but it lacks a "value" that I can measure in the usual way. It may take a bit of mental trickery to create a "placeholding marker" (like i for imaginary numbers) where division by zero occurs in a formula, with neat tricks for working it out of the formula again. So division by zero wouldn't necessarily make a formula unsolvable any more.

Zero, in my mind, has both a place (at the origin on a scale) and a value (distance measured from the origin along a dimension). Division by zero, however, has only a place (related to the origin) but no value. It has lost all dimensionality so it can't have a "value" in the usual sense. That information is lost.

question finally: So does this metaphor sound familiar to anyone? Could you point me in the direction of a book or an author (mathematician?) that describes stuff like this? Has somebody already explored the concept of Lost numbers? I'd like to learn more.

Thinking about my nickname and its age, I decide to google myself again to see what kind of trail I've left. Now that I participate in LiveJournal, I had to exclude that site to weed out many of the hits. I did find a post to an email list back in 1989. What's funny is that it was back in the days before the internet, and I was posting on BITnet... yet it still ended up archived on the internet. *laugh*

Curiously, though, a similar but older (by a few days) post that google used to locate is no longer indexed by the search engine. I didn't realize that results were eventually weeded out like that. Very odd.

Back on 2003 Dec 18 Thu (I know because I was emailing folk about my discovery), I came to the realization that we potentially have evidence already that our entire universe is but a virtual simulation. I was learning graphics processing in Java and encountered the concept of "culling". With culling, objects are defined mathematically and then drawn visually on the screen only as needed. Objects that would be hidden by other foreground items are simply not drawn in the first place so that the computer saves on processing time needed. Why spend time drawing something that isn't ultimately needed, right?

I suddenly realized that this process works just like wave/particle duality in physics, where something exists as a wave (when it has the potential to exist) until it collapses into a particle at observation (when it's really needed to exist somewhere). As I see it (pun intended), "wave function collapse" is just "back-face culling" in action.

So Slashdot hosted a discussion last week on the topic of our universe as a simulation based on a recent article titled, "The Physical World as a Virtual Reality". I didn't quite follow some of that paper, but it seemed to me that my culling example was not presented in the article. So I wrote the author to let him know of my insight a few years ago. He wrote back this week to explain that he did include the idea. "If what we call reality is a multi-dimensional space-time interface, it would likewise be expected to be calculated only on demand. " But he thanked me for introducing him to the term "culling" which he had not heard previously.

So, there it is. I did my (teeny, tiny, insignificant) part to expand our understanding that this universe doesn't... well... exist. Sort of. :)

I noticed the warning a few days ago that our primary supply of helium is nearly exhausted. The world's largest reservoir of it is located in the Texas panhandle (near Amarillo) and will likely run dry around 2016. The article explained how helium is produced on Earth, but it also stated, "... any helium ultimately released into the atmosphere by users, drifts up and is eventually lost to the Earth." This statement contradicts my own calculations on the matter, so naturally I was curious to learn the details.

Near as I can tell, my original calculations are appropriate but they should be applied in more specialized circumstances. It is not, apparently, the temperature at the surface that matters for giving molecules their kinetic energy for escape velocity. Our atmosphere has different layers, and it is the temperature in the highest layer, the exosphere, that determines the possibility of thermal escape for various molecules of gas. It's 1800 degrees in the exosphere, a lot hotter than down here on the surface, and that's plenty of energy for sending atoms or even whole molecules zooming off into space.

Aside: I didn't know before, but this process is called the Jean mechanism, though I can't find why it has that name. There is also the possibility of electric escape, with ions snatched away by solar flux. On Venus, in contrast, it is electric repulsion of atoms from the atmosphere itself after just the electrons are lost via thermal escape.

But then which molecules at the surface manage to make it up to the exosphere? I don't know. Knowing that information is necessary in order to make my calculations again. Apparently Earth does lose more atmosphere than I originally thought. I wish I knew better how to calculate that loss.

Back to helium though.... apparently helium exists near equilibrium these days (being produced in the Earth and lost to space at equal rates). But with the convenient ground sources nearly exhausted, and with important scientific and medical uses for helium taking precedence, it may be only a few years before party balloons disappear unless we find another safe alternative. I sure can't imagine anyone selling hydrogen balloons. (Mini-Kaboom!) Helium is already more expensive (per liquid gallon) than gasoline. It would get a lot more expensive if producers have to harvest it from the atmosphere directly.